On conflict-free proper colourings of graphs without small degree vertices

A proper vertex colouring of a graph G is conflict-free if in the neighbourhood of every vertex some colour appears exactly once, while it is called h-conflict-free if there are at least h such colours for each vertex of G. The least numbers of colours in such colourings of G are denoted by chi(pcf)(G) and chi(pcf)h(G), respectively. The latter parameter may be regarded as a natural relaxation of the 2-chromatic number, chi(2)(G), i.e. the least number of colours in a proper colouring of the square of a given graph G. It is known that chi(h)(pcf)(G) can be as large as (h+1)(Delta+1)approximate to Delta(2) for graphs with maximum degree Delta and h very close to Delta. We provide several new upper bounds for these parameters for graphs with minimum degrees delta large enough and h of smaller order than delta. In particular, we show that chi(pcf)h(G) <= (1+o(1))Delta if delta >> ln Delta and h << delta, and that chi(pcf)(G <= Delta+O(ln Delta) for regular graphs. These are related with the conjecture of Caro, Petrusevski and Skrekovski that chi(pcf)(G) <= Delta+1 for every connected graph G of maximum degree Delta >= 3, towards which they proved that chi(pcf)(G) <= [5 Delta/2] if Delta >= 1. (c) 2023 The Author(s). Published by Elsevier B.V.